Cartesian Product of Group Actions
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Theorem
Let $\struct {G, \circ}$ be a group.
Let $S$ and $T$ be sets.
Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions.
Then the operation $*: G \times \paren {S \times T} \to S \times T$ defined as:
- $\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$
is a group action.
Proof
The group action axioms are investigated in turn.
Let $g, h \in G$ and $s, t \in S$.
Thus:
\(\ds g * \tuple {h * \tuple {s, t} }\) | \(=\) | \(\ds g * \tuple {h *_S s, h *_T t}\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {g *_S \tuple {h *_S s}, g *_T \tuple {h *_T t} }\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {\paren {g \circ h} *_S s, \paren {g \circ h} *_T t}\) | Group Action Axiom $\text {GA} 1$ for both $*_S$ and $*_T$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {g \circ h} * \tuple {s, t}\) | Definition of $*$ |
demonstrating that Group Action Axiom $\text {GA} 1$ holds.
Then:
\(\ds e * \tuple {s, t}\) | \(=\) | \(\ds \tuple {e *_S s, e *_T t}\) | Definition of $*$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {s, t}\) | Group Action Axiom $\text {GA} 2$ for both $*_S$ and $*_T$ |
demonstrating that Group Action Axiom $\text {GA} 2$ holds.
The group action axioms are thus seen to be fulfilled, and so $*$ is a group action.
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Exercise $3$